Let S=[-π, π/2)-{-π/2,-π/4,-(3 π)/4, π/4}. Then the number of elements in the set | A={θ ∈ S: tan θ(1+√5 tan (2 θ))=√5-tan (2 θ)} is __________.

Solution

Maths sir · 4 steps

Group the terms into the tan(θ + 2θ) formula, then solve tan 3θ = √5.

1. Rearrange. Expanding gives tan θ + √5 tan θ tan 2θ = √5 - tan 2θ. Group the terms as tan θ + tan 2θ = √5 (1 - tan θ tan 2θ).

   (tan θ + tan 2θ)/(1 - tan θ tan 2θ) = tan 3θ = √5

2. Division is safe. If 1 - tan θ tan 2θ = 0, the equation forces tan θ = -tan 2θ, so tan θ tan 2θ = -tan² θ ≤ 0, which cannot equal 1. So dividing loses no root. The points where tan θ or tan 2θ is undefined are -π/2, ±π/4, -3π/4, and they are already removed from S.
3. General solution. Let α = tan⁻¹√5 ≈ 65.9°. Then 3θ = α + nπ. Since θ ∈ [-π, π/2), we need 3θ ∈ [-3π, 3π/2), which allows n = -3, -2, -1, 0, 1.

   n	θ = (α + nπ)/3	In S?
   -4	≈ -218.0°	no, below -180°
   -3	≈ -158.0°	yes
   -2	≈ -98.0°	yes
   -1	≈ -38.0°	yes
   0	≈ 22.0°	yes
   1	≈ 82.0°	yes
   2	≈ 142.0°	no, beyond 90°

4. So. None of the five values is an excluded point, so A has 5 elements.

The idea to keep. When you see tan A + tan B next to 1 - tan A tan B, think compound angle at once. Then convert the θ interval into the interval for 3θ before you count values of n.

Answer from the NTA final answer key. Solution by PrepWiser.